ON MODULES WITH LINEAR PRESENTATIONS
E. L. GREEN, R. MARTINEZ-VILLA, I. REITEN, Ø. SOLBERG, AND D. ZACHARIA
Introduction
Let Λ = Λ0 +Λ1 +· · · be a graded algebra generated in degree 0 and 1 over a field
k, such that each Λi is a finite dimensional vector space over k and Λ0 is a finite
product of copies of k. Denote by gr Λ the category of finitely generated graded
Λ-modules with degree zero morphisms. Recall that Λ is Koszul if Λ0 has a linear
resolution, that is, there is a projective resolution · · · → Pi → · · · → P1 → P0 →
Λ0 → 0 of Λ0 in gr Λ, with each Pi generated in degree i. We denote by K(Λ) the
subcategory of gr Λ whose objects have such a projective resolution. Koszul algebras
have been studied a lot in recent years, see for example [BGS], [GM1], [GM2], [GZ]
and [Sm]. Associated with Λ is the Yoneda Ext-algebra E(Λ) = Ext∗Λ (Λ0 , Λ0 ), and
when Λ is Koszul there is a natural duality between K(Λ) and K(E(Λ)op ). An
interesting feature of this duality is that it sometimes relates module categories
over infinite dimensional algebras with module categories over finite dimensional
algebras.
There are interesting subcategories of gr Λ other than K(Λ), defined in the same
spirit, by considering for a given n ≥ 0 the module C having a projective resolution
· · · → Pi → · · · → P1 → P0 → C → 0 such that Pi is generated in degree i for i ≤ n.
We consider especially the cases n equals 0 or 1. For n = 0 we get the category gr0 Λ
of modules in gr Λ generated in degree zero. If n = 1 the modules are said to have a
linear presentation, and we denote the corresponding subcategory by L(Λ). For the
categories K(Λ), L(Λ) and gr0 Λ we investigate to which extent they have almost
split sequences. This is an interesting question about subcategories from the point
of view of the representation theory of artin algebras. We prove that gr0 Λ and
L(Λ) have almost split sequences when Λ is a finite dimensional k-algebra, and at
the same time we give a description of the projective and injective objects, which is
also an interesting problem for subcategories. For K(Λ) there is not necessarily an
almost split sequence 0 → A → B → C → 0 when A is an indecomposable module
in K(Λ) which is not injective. Such examples are found when Λ is selfinjective
Koszul of Loewy length greater than 3. However, given C indecomposable and
not projective in K(Λ), there is an almost split sequence 0 → A → B → C → 0
in the selfinjective case, but we do not know if this is true in general, even for
Koszul algebras. In this connection we give some new sufficient conditions on C
for the middle term B in the almost split sequence 0 → A → B → C → 0 to be
indecomposable.
Part of this research was completed under the US-Mexico exchange grant of the NSFCONACYT. Green and Zacharia would like to thank the American-Scandinavian Foundation
for support through the Maurice Auslander award, and Martinez-Villa would like to thank the
Norwegian University of Science and Technology for their partial support of this research. Reiten
and Solberg would like to thank the Norwegian research council for partial support.
1
2
GREEN, MARTINEZ-VILLA, REITEN, SOLBERG, AND ZACHARIA
The category L(Λ) is easier to handle than the usually smaller category K(Λ).
For the study of K(Λ) it is hence of interest to investigate when they coincide. We
prove in the first section that this is the case for quadratic monomial algebras, and
give an example to show that it may hold also when the algebra is not monomial.
For the study of L(Λ) it is useful that there is an equivalence L(Λ) ≈ L(Λ/(rad Λ) 2 ),
and hence L(Λ) ≈ gr0 (Λ/(rad Λ)2 ).
The finitistic dimension conjecture, which states that there should be a bound
on the projective dimension of modules of finite projective dimension, is one of the
most intriguing problems for artin algebras. It is also interesting to know if the
conclusion is true when restricting to various types of subcategories, and here we
show that it holds for Koszul modules over finite dimensional k-algebras.
We thank the referee for very helpful comments.
1. Linear presentations
In this section we start the study of the modules with linear presentations over
a graded algebra. We also introduce the notation and terminology that is needed
throughout this paper.
`
Let k be a field and let Λ = i≥0 Λi be a graded k-algebra. We recall that
Λ is said to be generated in degree 0 and 1, if for each i, j ≥ 0 we have Λi Λj =
Λi+j . These algebras have also been called homogeneous, and also strongly graded.
Throughout this paper, a graded k-algebra Λ will always denote a graded k-algebra
generated in degree 0 and 1, such that, for each i ≥ 1, Λi is a finite dimensional
k-vector space, and that Λ0 is isomorphic to a finite direct product of the ground
field. Such an algebra is the quotient of the tensor algebra over Λ0 of Λ1 by an ideal
generated by homogeneous tensors of degrees 2 or more, and therefore we can write
Λ as a quotient of the path algebra kQ of a finite quiver Q modulo a two-sided
ideal I satisfying the following conditions:
(a) I ⊆ J 2 where J is the two-sided ideal of the path algebra kQ generated by
the arrows of Q.
(b) I is generated by length
Pn homogeneous elements, that is, the generators of I
are elements of the form i=1 αi pi where n ≥ 1, αi are nonzero scalars and the
paths p1 , . . . , pn have the same origin, the same terminus, and the same length
(which must be at least two by part (a)).
As usual, r will denote the graded radical of Λ. (For instance if Λ = kQ, the path
algebra of a quiver Q, then r = J, the ideal generated by the arrows.) It is clear that
if Λ is finite dimensional then r is the Jacobson radical of the algebra. We denote
by gr Λ the category of finitely generated graded left Λ-modules and degree zero
homomorphisms. A graded Λ-module M = Mj + Mj+1 + · · · is said to be generated
in degree j, if for each i ≥ 0, Mi+j = Λi Mj . For instance if Λ = Λ0 + Λ1 + · · · is
generated in degree 0 and 1, then as a graded module Λ is generated in degree 0.
We denote by gr0 Λ the full subcategory of gr Λ consisting of the graded modules
generated in degree 0.
We say that a graded module M = M0 + M1 + · · · in gr0 Λ has a linear presentation if there is an exact sequence P1 → P0 → M → 0 in gr Λ, where P0
and P1 are projective modules generated in degree 0 and 1 respectively. We denote by L(Λ) the full subcategory of gr0 Λ consisting of the modules having linear presentations. We say that a graded module M = M0 + M1 + · · · in gr0 Λ
has a linear resolution, or is a Koszul module, if there is a projective resolution
ON MODULES WITH LINEAR PRESENTATIONS
fn
3
f1
· · · → Pn −→ Pn−1 → · · · → P1 −→ P0 → M → 0 of M in gr Λ such that for each
i we have that Pi is generated in degree i. It is easy to see that such a resolution
must be minimal in the sense that for each i ≥ 0 we have Im fi ⊆ rPi−1 . We
denote by K(Λ) the full subcategory of gr0 Λ consisting of those modules having
linear resolutions. Obviously K(Λ) ⊆ L(Λ). We recall that the graded algebra Λ
is Koszul if Λ0 = Λ/r ∈ K(Λ).
This is equivalent (see [BGS], [GM2]) to the fact
`
that the Yoneda algebra n≥0 ExtnΛ (Λ0 , Λ0 ) is generated in degree 0 and 1. (In
`
other words Λ! = n≥0 ExtnΛ (Λ0 , Λ0 ) where Λ! is the shriek algebra of Λ as defined
in [BGS]). It is known ([BGS]) that every Koszul algebra is quadratic, that is, if
Λ = kQ/I where Q is a finite quiver and I ⊆ J 2 , then I is generated by elements
of kQ that are k-linear combinations of paths of length two. In the case that Λ is
monomial, that is I is generated by paths in Q, then Λ is a Koszul algebra if and
only if it is quadratic [GZ].
We will need the following observation.
Lemma 1.1. Let P be an indecomposable projective Λ-module. Then, for each j ≥
0 we have isomorphisms of Λ0 -modules Λj ⊗Λ0 P/rP ' r j P/rj+1 P . In particular
we have Λ1 ⊗Λ0 P/rP ' rP/r 2 P . Furthermore, if P = Pi is a projective Λ-module
which is generated in degree i, then we have Pi,i = Pi /rPi , and, if j > i, its j-th
component Pi,j can be written as Pi,j = Λj−i ⊗Λ0 Pi /rPi = qp ⊗Λ0 Pi /rPi where p
runs through a k-basis of the right Λ0 -module Λj−i .
Proof. Since P is projective we have r j ⊗Λ P ' r j P for each j ≥ 0, therefore
also rj /r j+1 ⊗Λ P ' r j P/rj+1 P . Now since Λj = rj /r j+1 and Λ0 = Λ/r we
have Λj ⊗Λ P ' r j P/rj+1 P and also r j /rj+1 ⊗Λ P ' (r j /rj+1 ⊗Λ/r Λ/r) ⊗Λ P '
rj /r j+1 ⊗Λ0 (Λ/r ⊗Λ P ) ' r j /rj+1 ⊗Λ0 P/rP . The remaining statements follow
immediately.
Let M be in L(Λ), and let
f
P1 −
→ P0 → M → 0
be a linear presentation of M . Then, in degree 1, we have a monomorphism of
Λ0 -modules
(1)
f1
P1 /rP1 −→ Λ1 ⊗Λ0 P0 /rP0 ' rP0 /r 2 P0
which induces f uniquely in gr Λ:
f : P1 = Λ ⊗Λ0 P1 /rP1 → P0 = Λ ⊗Λ0 P0 /rP0
by f (λ ⊗ x) = λf1 (x).
For the rest of this paper all graded algebras Λ = Λ0 + Λ1 + · · · will be homogeneous, that is, they will be generated in degree 0 and 1. We study next a reduction
procedure that facilitates the study of modules with linear presentation.
Proposition 1.2. Let Λ be a graded algebra as above. There is an exact equivalence
between the categories L(Λ) and L(Λ/r 2 ).
f
Proof. Consider first the category whose objects are degree zero maps P 1 −
→ P0
with P0 projective generated in degree 0 and P1 projective generated in degree 1,
such that the degree one component of f takes (P1 )1 monomorphically into (P0 )1 ,
4
GREEN, MARTINEZ-VILLA, REITEN, SOLBERG, AND ZACHARIA
f
g
and where the morphisms between objects P1 −
→ P0 and Q1 −
→ Q0 are pairs of
degree zero homomorphisms (α1 , α0 ) making the following diagram commutative
P1
f
α1
Q1
/ P0
α0
g
/ Q0
A slight modification of the proof of [ARS, Prop. 1.2, page 102] shows that L(Λ)
is equivalent to this category, and we will identify L(Λ) with this category. We
f
now define functors F : L(Λ) → L(Λ/r 2 ) and G : L(Λ/r2 ) → L(Λ). If P1 −
→ P0
is a degree zero homomorphism of projective Λ-modules such that P1 → P0 →
f
f
→ P 0 ) where
Coker f → 0 is a linear presentation, we put F (P1 −
→ P0 ) = (P 1 −
P i = Pi /r2 Pi for i = 0, 1, and f is the induced degree zero map. (Note that if
P0 = (P0 )0 +(P0 )1 +· · · , then P 0 = (P0 )0 +(P0 )1 , and similarly P 1 = (P1 )1 +(P1 )2 ).
f
Vice versa, let P 1 −
→ P 0 be a degree zero homomorphism of projective Λ/r 2 modules such that P 1 → P 0 → Coker f is a linear presentation. We can lift
P 0 and P 1 to two projective Λ-modules P0 and P1 by taking (P0 )0 = (P 0 )0 and
then (P0 )i = Λi (P0 )0 for each i > 0, and similarly for P1 . As we have seen, the
map f also lifts to a degree zero map f : P1 → P0 in the obvious way by defining
f
f
→ P 0 ) = P1 −
f1 : (P1 )1 → (P0 )1 to be (f )1 . Then we put G(P 1 −
→ P0 . The
definition of F and G on morphisms is obvious, and it is clear that F and G are
exact equivalences inverse to each other.
It is easy to see that L(Λ/r 2 ) = gr0 (Λ/r2 ). We have the following immediate
consequence.
Corollary 1.3. Let Λ be a graded algebra and let M ∈ L(Λ). Then M is indecomposable over Λ if and only if M/r 2 M is indecomposable over Λ.
We have the following result about the representation type of L(Λ) by reducing
to a finite dimensional radical squared zero algebra. We use the fact that over such
an algebra every finitely generated module is gradable, and we obtain:
Corollary 1.4. Let Λ be a graded algebra. Then the category L(Λ) is of finite
(tame, wild) representation type if and only if Λ/r 2 is of finite (tame, wild) representation type.
In view of 1.2 and 1.4 it is of interest to investigate the algebras with radical
squared zero. More generally, the monomial algebras are important in this connection, especially because of the following.
Theorem 1.5. Let Λ be a quadratic monomial algebra. Then L(Λ) = K(Λ).
f
Proof. Let P1 −
→ P0 → M → 0 be a linear presentation of M with P0 generated in
degree 0, and P1 generated in degree 1. To show that M is in K(Λ) it suffices to
prove that Ker f is generated in degree 2. From 1.1 we know that we can write the
s component of P1 as:
P1,s = Λs−1 ⊗Λ0 P1 /rP1 = qp ⊗Λ0 P1 /rP1
ON MODULES WITH LINEAR PRESENTATIONS
5
where, since Λ is monomial, p runs over all the paths of length s − 1 which
P are
nonzero in Λ. Let z be an element of degree s of Ker f , so
we
can
write
z
=
p⊗xp
P
where the xp are in P1 /rP1 . But f (z) = 0 implies that
pf1 (xp ) = 0. The algebra
Λ being monomial, we infer now that each p⊗xp is in Ker f , so we may assume that
z = p ⊗ x where x is the primitive
idempotent corresponding to the terminus of p.
P
Using (1) we write f1 (x) = ca a ⊗ va where a runs over a subset of the arrow set,
va is the primitive idempotent corresponding to the terminus of a, and we clearly
may assume that each coefficient ca is nonzero. Since f1 is a monomorphism, we
see that we may assume that xa = a for each a. Since f (z) = 0 we have:
X
pf1 (x) =
ca pa ⊗ ca = 0
where the sum runs over all the arrows a with ca 6= 0, such that p and a are
composable. We infer then that pa = 0 in Λ for all such arrows. Since Λ is
quadratic we can write p = p0 b where p0 is a nonzero path in Λ of length s − 2, and
b is an arrow such that ba = 0 in Λ for all the arrows a above, since Λ is quadratic.
Let z 0 = b ⊗ x. Then z 0 is in the degree 2 part of P1 , z 0 is also in Ker f , and,
z = p0 z 0 . Therefore Ker f is generated in degree 2.
Note that since we have the linear presentation Λ ⊗Λ0 Λ1 → Λ → Λ0 → 0, the
module Λ0 is always in L(Λ). Hence, if L(Λ) = K(Λ), it is necessary that Λ is
Koszul. But the following examples show that Λ being Koszul is not a sufficient
condition for L(Λ) = K(Λ), and that monomial Koszul is not a necessary condition.
It is not known for which algebras we have L(Λ) = K(Λ).
Example. We consider the commutative algebra Λ = k[x, y]/(x2 , y 2 ). It is easy
to verify that Λ is a Koszul algebra. To show that K(Λ) 6= L(Λ) we construct a
(x,y)
module in L(Λ) as follows. Let M be the cokernel of the map φ : Λ −−−→ Λ q Λ.
This gives a linear presentation of M , so that M is in L(Λ). An easy computation
gives Ker φ = {a0 + a1 y + a2 x + a3 x y ∈ Λ | a0 x + a1 x y = a0 y + a2 x y = 0} =
{a3 x y} = soc Λ = r 2 . So Ker φ is generated in degree three and hence M is not a
Koszul module.
4
•
>> d
>>

•3 subject to the
Example. Let Λ be the algebra given by the quiver 2• >
>>
>
a  c
•
b
1
relation ab = cd. Since Λ is quadratic and of global dimension two it must be
Koszul [GM1]. Since the algebra Λ is of finite representation type, it is easy to
show directly that K(Λ) = L(Λ). The algebra Λ is clearly not monomial.
We have the following consequence 1.2 and 1.5 which also enables us to describe
the Koszul modules over a quadratic monomial algebra.
Proposition 1.6. Let Q be a finite quiver and let I1 and I2 be two-sided ideals of
the path algebra kQ such that they are both length homogeneous and are contained
in J 2 where J is the ideal of kQ generated by the arrows. Then we have an exact
equivalence between L(kQ/I1 ) and L(kQ/I2 ). In particular if Λ is a quadratic
monomial algebra we have an exact equivalence between K(Λ) and L(Λ/r 2 ).
Corollary 1.7. Let Λ be a graded algebra. Then we have a natural equivalence
between L(Λ) and K(Λ/r 2 ) = L(Λ) = gr0 (Λ/r2 ).
6
GREEN, MARTINEZ-VILLA, REITEN, SOLBERG, AND ZACHARIA
We shall see shortly that given two length homogeneous ideals I1 and I2 as above,
we also have a duality between L(kQ/I1 ) and L(kQ/I2 ). But first we give some
examples. The first one illustrates how to use 1.2 and 1.5 to find all Koszul modules
directly even for a wild algebra. The second example shows that the representation
type of L(Λ) (or K(Λ)) is not preserved under tilting, not even under change of
orientation for hereditary algebras.
Example. Let Λ be the hereditary algebra Λ = kQ where Q is the quiver
4
5
• ::
•
::
::
::
•
3
•
1
6
•
::
::
::
::
•
2
Let us compute the indecomposable Koszul modules. We could try to use the
result of [GM2] that says that an indecomposable module over a hereditary algebra
is Koszul if and only if its radical is projective, but this would prove rather difficult
since our algebra is of wild representation type. However, since it is hereditary, it
is also monomial, so K(Λ) ≈ L(Λ) ≈ L(Λ/r 2 ) = gr0 Λ/r2 . Now Λ/r2 is of finite
representation type so it is easy to find the indecomposable modules in gr0 Λ/r2 .
Applying the equivalence from Proposition 1.2 we get that the indecomposable
Koszul Λ-modules correspond to the following representations:
S(1), S(2), S(3), S(4), S(5), S(6), P (3), P (4), P (5), P (6), 31 ,
4 5 6
32
22
12
,
5
1
3
6
2
,
4
1
3
6
2
,
4
1
3
5
2
,
3
2
45 6
3 .
1 2
where S(1)–S(6) are the simple modules corresponding to the vertices 1–6 and
P (1)–P (6) are their projective covers. (Note that S(1) = P (1) and S(2) = P (2)).
Example. Let Γ = kQ0 where Q0 is the quiver
•@
•
•
•
@@
 ooooo

@@
 o
@@ ooooo
wooo
•
•
Then Γ is tilted from Λ using a partial Coxeter functor and L(Γ) ≈ L(Γ/r 2 Γ) is of
infinite (tame) representation type.
`
We now assume that Λ is a Koszul algebra. Let E(Λ) = n≥0 ExtnΛ (Λ0 , Λ0 ) be
the Yoneda Ext-algebra, and let A be the full subcategory of gr0 E(Λ)op whose objects are the graded modules of the form X = X0 +X1 , so A = gr0 (E(Λ)op /r2E(Λ)op ).
We now show that there is a duality between L(Λ) and the category A. We want
to give an explicit description of this duality and its inverse. Note that this duality
does not extend the usual Koszul duality.
ON MODULES WITH LINEAR PRESENTATIONS
7
If M ∈ L(Λ) we define H(M ) = Ext0Λ (M, Λ0 ) q Ext1Λ (M, Λ0 ) ∈ A, so we let
f
X0 = Ext0Λ (M, Λ0 ) and X1 = Ext1Λ (M, Λ0 ). Note that if P1 −
→ P0 → M →
0 is a linear presentation of M , then we have Ext0Λ (M, Λ0 ) = HomΛ (P0 , Λ0 ) =
HomΛ (P0 /rP0 , Λ0 ) = HomΛ0 (P0 /rP0 , Λ0 ) = (P0 /rP0 )∗ where we denote by A∗ the
Λop
0 -module HomΛ0 (A, Λ0 ) for every Λ0 -module A. Similarly X1 = HomΛ (P1 , Λ0 ) =
(P1 /rP1 )∗ . In particular we get E(Λ)0 = Λ∗0 = Λ0 and E(Λ)1 = Λ∗1 .
We show next that the module H(M ) = X = X0 + X1 is generated in degree 0.
f
Since P1 −
→ P0 → M → 0 is a linear presentation of M we have a monomorphism
of Λ0 -modules
(f )1
P1 /rP1 −−→ rP0 /r2 P0 ' Λ1 ⊗Λ0 P0 /rP0 .
By taking duals we obtain the following commutative diagram with exact rows:
(Λ1 ⊗Λ0 P0 /rP0 )
(∗)
∗
o
∗
(P0 /rP0 ) ⊗Λ0 Λ∗1
X0 ⊗E(Λ)0 E(Λ)1
/ (P1 /rP1 )∗
/0
/ (P1 /rP1 )∗
/0
/ X1
/0
ξ
where the first vertical isomorphism is due to the fact that for all finitely generated
Λ0 -modules A and B we have (A⊗Λ0 B)∗ ' B ∗ ⊗Λ0 A∗ and ξ is the induced map. In
order to conclude that X = X0 +X1 is generated in degree 0 as a right E(Λ)-module
it is enough to prove the following:
Lemma 1.8. The usual E(Λ)op -module structure on H(M ) = X coincides with
the structure given by the map ξ in (∗).
e be in
Proof. Let the notation be as above. Let qe be in X0 = (P0 /rP0 , Λ0 ), and let λ
E(Λ)1 = (r/r2 , Λ0 ). Then consider the following commutative diagram with exact
horizontal rows:
P1 B
BB
BBf
BB
B!
/ rP0
0
II
II
II
II
I$
qe1
rP0 /r 2 P0
qe2
0
/rI
II
II
II
II
I$
r/r 2
e
λ
Λ0
/ P0
qe1
/Λ
/ P0 /rP0
/0
qe
/ Λ0
/0
8
GREEN, MARTINEZ-VILLA, REITEN, SOLBERG, AND ZACHARIA
where qe1 is a lifting of qe to P0 and qe2 is the induced map. Recall that the usual
e = pe where pe: P1 /rP1 → Λ0 is the map pe(b
module structure is given by qeλ
x) =
e
λe
q2 (f (x)) for x
b ∈ P1 /rP1 (∗∗).
On the other hand, the module structure on X given by the map ξ in (∗) is given
by the composition
∗
(P0 /rP0 ) ⊗Λ0 Λ∗1
φ
∼
/ (Λ1 ⊗Λ P0 /rP0 )∗
0
(ψ◦fb)∗
/ (P1 /rP1 )∗
e
e
⊗ y) = qe(y)λ(r),
the map ψ : rP0 /r2 P0 → Λ1 ⊗Λ0
where φ is given by φ(e
q ⊗ λ)(r
P
P
P0 /rP0 is the isomorphism given by ψ( ri yi ) =
r i ⊗ yi , and fb: P1 /rP1 →
f
rP0 /r2 P0 is the map induced by the composition P1 −
→ rP0 → rP0 /r2 P0 . It is easy
∗
b
e
to show that (ψ ◦ f ) (φ(e
q ⊗ λ)) = pe where pe is as defined above (∗∗). The lemma
follows from these considerations.
We now construct an inverse to the functor H. Let X = X0 + X1 ∈ A. We have
an exact sequence in gr E(Λ)op : 0 → X1 → X → X0 → 0 where X0 and X1 are
semisimple E(Λ)op -modules. Applying HomE(Λ) ( , Λ0 ) to this sequence we obtain
(∗)
(g)1
HomE(Λ) (X0 , Λ0 ) → HomE(Λ) (X, Λ0 ) → HomE(Λ) (X1 , Λ0 ) −−→
(g)2
Ext1E(Λ) (X0 , Λ0 ) → Ext1E(Λ) (X, Λ0 ) → Ext1E(Λ) (X1 , Λ0 ) −−→ Ext2E(Λ) (X0 , Λ0 ) → · · ·
where the map HomE(Λ) (X, Λ0 ) → HomE(Λ) (X1 , Λ0 ) is zero since X is generated in degree zero. Thus X1 is the radical of `
X (X1 = Xr). Therei
fore the map (g)1 is a monomorphism. Let Q0 =
i≥0 ExtE(Λ) (X0 , Λ0 ) and
`
Q1 = i≥0 ExtiE(Λ) (X1 , Λ0 )[1]. Since X0 and X1 are semisimple over E(Λ) (and
E(Λ) is Koszul), Q0 and Q1 are both projective E(E(Λ))-modules generated in
degree 0 and 1, respectively. The exact sequence in (∗) tells us that there is a map
g : Q1 → Q0 giving a linear presentation of Coker g. We now define H : A → L(Λ)
by putting H(X) = Coker g. Note that we use the fact that Λ is Koszul in order to get L(E(E(Λ))) ≈ L(Λ). An easy computation shows that H and H are
dualities inverse to each other and that they both take simple modules to indecomposable projective modules, and projective modules to simple modules. We have
then proven:
Proposition 1.9. Let Λ be a Koszul algebra. The functors H : L(Λ) → A and
H : A → L(Λ) defined above are inverse dualities.
As an immediate consequence, since kQ is a Koszul algebra, we obtain by combining the results of 1.2 and 1.9 the following:
Theorem 1.10. Let Q be a finite quiver and let I1 and I2 be two length homogeneous ideals of kQ both contained in J 2 . Then there exist dualities between
L(kQ/I1 ) and L(kQ/I2 ) taking simple modules into indecomposable projective modules and indecomposable projective modules into simple modules.
Proof. One only has to observe that E(kQ) = kQ/J 2 so we have duality between
L(kQ) and L(kQ/J 2 ). We then apply 1.2 and 1.9.
ON MODULES WITH LINEAR PRESENTATIONS
9
2. The subcategories gr0 Λ and L(Λ)
From now on we assume that Λ is a finite dimensional k-algebra. In this section
we determine the projective and the injective objects in gr0 Λ and in L(Λ), and we
also show that both gr0 Λ and L(Λ) have almost split sequences. We observe that
both gr0 Λ and L(Λ) are closed under extensions and contain the projective and
the semisimple modules generated in degree zero.
We first recall some results due to Auslander and Smalø [ASm], here formulated
for full subcategories of gr Λ. The original proofs can be modified.
Theorem 2.1. Let T be a torsion class in gr Λ, that is, a full subcategory of gr Λ
closed under factor modules, extensions and isomorphisms. If a module X is in
gr Λ, let tX denote the trace of T in X, that is the unique submodule of X maximal
with respect to being in T . Let M ∈ T be an indecomposable module.
(a) M is an injective object in T if and only if there exists an indecomposable
injective module I with M ' tI.
(b) M is a projective object in T if and only if t(D Tr M ) = 0.
(c) T is contravariantly finite in gr Λ.
(d) If T is the set Fac T of all factor modules of all finite direct sums of copies
of a fixed module T in T , then T is covariantly finite in gr Λ, and T has almost
split sequences.
To see that T is contravariantly finite in gr Λ one only needs to observe that for
C in gr Λ the inclusion map t(C) → C is a right T -approximation. To see that T =
Fac T is covariantly finite in gr Λ one first constructs a left add T -approximation
of the projective cover of a given module and then take the pushout of the left
add T -approximation and the projective cover.
We can modify [ASm, 2.4] to obtain the existence of almost split sequences for
T = Fac T . We sketch the proof for the reader’s convenience: Consider for C an
indecomposable nonprojective object in T the almost split sequence 0 → D Tr C →
B → C → 0 in gr Λ. Then the sequence 0 → t(D Tr C) → t(B) → C → 0 is proved
to be exact, and some summand of this sequence is an almost split sequence in T
with right hand term C. If A is indecomposable noninjective in T , let 0 → A →
B → C → 0 be almost split in gr Λ. Let B → E be a left T -approximation. Then
the composition g : A → E is a monomorphism which is left almost split in T , and
Coker g is clearly in T . So a summand of 0 → A → E → Coker g → 0 is an almost
split sequence in T with A as left hand term.
Clearly the subcategory T = gr0 Λ of gr Λ is a torsion class. Then we have
tM = hM0 i for M in gr Λ. Using T = Fac Λ we obtain the following consequence
of 2.1.
Theorem 2.2. The category gr0 Λ has almost split sequences.
In order to describe the indecomposable modules in gr0 Λ which occur as end
terms of almost split sequences, we need to find the injective and projective objects
in gr0 Λ.
Proposition 2.3. Let M ∈ gr0 Λ be an indecomposable module. The following are
equivalent:
(a) M is an injective object in gr0 Λ.
(b) There exists an indecomposable injective module I in gr Λ such that M = hI 0 i.
10
GREEN, MARTINEZ-VILLA, REITEN, SOLBERG, AND ZACHARIA
(c) There exists an indecomposable injective module I and a positive integer n
such that M ' h(socn I)0 i.
Proof. (a) and (b) are equivalent by the preceding remarks and the equivalence
of (b) and (c) is obvious.
Proposition 2.4. Let M ∈ gr0 Λ be an indecomposable module. Then M is a
projective object in gr0 Λ if and only if (D Tr M )0 = 0.
We want to give a more explicit description of the projective objects in gr0 Λ.
Also we want to show that if C is indecomposable nonprojective in gr0 Λ and
0 → D Tr C → B → C → 0 is almost split in gr Λ, then 0 → h(D Tr C)0 i → hB0 i →
C → 0 is almost split in gr0 Λ. In other words, we do not need to take a (proper)
summand.
Lemma 2.5. Let M be an indecomposable nonprojective module in gr Λ generated
in degree zero. The following statements are equivalent:
(i) M is a projective object in gr0 Λ.
g
(ii) There exists a short exact sequence 0 → S → B −
→ M → 0 where g is right
almost split in gr0 Λ and S is a simple module generated in degree greater than zero.
g
(iii) There exists a short exact sequence 0 → K → B −
→ M → 0 where g is right
almost split in gr0 Λ and K is generated in positive degree.
g
→ M → 0 be right almost split in gr Λ. Consider the
Proof. (i) ⇒ (ii). Let E −
exact sequence 0 → K → hE0 i → M → 0, where hE0 i → M is right almost
g
i
split in gr0 Λ. There is an exact sequence 0 → K 0 −
→ B −
→ M → 0 in gr Λ,
00
where g is minimal right almost split in gr0 Λ and some K such that the sequence
0 → K 0 q K 00 → B q K 00 → M → 0 is isomorphic to 0 → K → hE0 i → M → 0.
Assume now that M is a projective object in gr0 Λ. We want to show that K 0 is
simple and generated in a positive degree. We first note that K 0 6∈ gr0 Λ since M
is Ext-projective in gr0 Λ. Hence we have K 0 6= hK00 i, and so K 0 /hK00 i is generated
in positive degree. Let S be a simple module in the top of K 0 /hK00 i. We have the
following pushout diagram:
0
/ K0
0
/S
i
/B
g
/W
/M
/0
/M
/0
and W is also in gr0 Λ since W is a factor module of B. The bottom sequence cannot
split since Homgr Λ (B, S) = 0, and it is then immediate that the map W → M is
right almost split in gr0 Λ. Since g was minimal right almost split it follows that
the vertical maps are isomorphisms i.e. K 0 ' S, B ' W and the implication is
proved.
(ii) ⇒ (iii). This is trivial.
(iii) ⇒ (i). Assume that M is not a projective object in gr0 Λand K generated
in positive degree. Then we have a commutative diagram with exact rows:
/X
/Y
/M
/0
0
0
/K
/B
g
/M
/0
ON MODULES WITH LINEAR PRESENTATIONS
11
where the top row is nonsplit with X and Y in gr0 Λ. But then we have
Homgr Λ (X, K) = 0, contradicting the fact that g does not split.
We also have the following necessary condition for an indecomposable nonprojective module M ∈ gr0 Λ to be a projective object in gr0 Λ.
Proposition 2.6. Let M be an indecomposable nonprojective module generated in
degree zero. If M is a projective object in gr0 Λ, then D Tr M has simple socle.
Proof. Assume that M is projective in gr0 Λ. By 2.5 we have a short exact sequence
g
i
of graded modules 0 → S −
→B−
→ M → 0 where g is right almost split in gr0 Λ and
S is a simple module generated in a positive degree. Let P ∈ gr0 Λ be the projective
cover of M and let L be its first syzygy. We have the following commutative diagram
with exact rows:
0
/L
0
/S
g
i
/P
i
/B
p
/M
/0
/M
/0
p
g
and the morphism p is nonzero since the bottom sequence does not split. Let
X = Ker p and let Y be a maximal submodule of L. We want to show that X = Y .
p0
Let p0 be the projection L −→ S 0 = L/Y . Consider the pushout
0
/L
0
/ S0
g
i
/P
j
/ W0
p0
/M
/0
/M
/0
p0
0
q
0
where W 0 ∈ gr0 Λ. If j : S 0 → W 0 was a split monomorphism, there would be a
nonzero map P → S 0 , which is impossible since S 0 is generated in positive degree.
Since g is right almost split in gr0 Λ and q is not a splittable epimorphism this gives
rise to the commutative diagram
0
/L
0
/ S0
0
/S
g
i
/P
j
/ W0
i
/B
p0
/M
/0
/M
/0
/M
/ 0.
p0
f
q
f
g
The morphism f cannot be zero since g does not split. Therefore both maps f and
f are isomorphisms. We have gfp0 = g but also gp = g. Therefore g(fp0 − p) = 0,
so there exists h : P → S such that ih = fp0 − p. But Homgr Λ (P, S) = 0, so h = 0
and then p = f p0 . This implies that pi = fp0 i = f jp0 = if p0 , and also ip = pi.
Thus i(p − f p0 ) = 0, therefore p = f p0 so Ker p0 ⊆ Ker p. This shows that Y ⊆ X,
but Y is a maximal submodule, so Y = X. We have proved that L/rL is simple.
But L/rL ' soc D Tr M [ARS], the proof is complete.
12
GREEN, MARTINEZ-VILLA, REITEN, SOLBERG, AND ZACHARIA
We shall need the following facts about graded modules. Let P be an indecomposable projective Λ-module generated in degree i. Then P ∗ = HomΛ (P, Λ) is an
indecomposable projective Λop -module generated in degree −i. We also know that
if M is a graded Λ-module, then its dual DM = Homk (M, k) is a graded Λop module with (DM )i = D(M−i ) for each i. Therefore if P is an indecomposable
projective Λ-module of Loewy length n generated in degree i, then P ∗ is generated
in degree −i, the socle of P ∗ is generated in degrees −i + n − 1 and lower, thus the
generators of DP ∗ are in degrees i − n + 1 and higher.
We can now get more information on the injective and projective objects in gr0 Λ.
Lemma 2.7. Let M be an indecomposable nonprojective module generated in degree zero such that M is projective in gr0 Λ. There exists an indecomposable injective Λ-module I and an integer n > 0 such that M ' Tr D(socn I) where socn I
denotes the n-th socle of I.
Proof. By 2.4 we have (D Tr M )0 = 0. Let P1 → P0 → M → 0 be a minimal projective presentation of M in gr Λ. By 2.6, the socle of D Tr M is simple,
so P1 is indecomposable generated in a positive degree. We have the exact sequence 0 → D Tr M → DP1∗ → DP0∗ . But (DP0∗ )j = 0 for all j > 0. Since
(D Tr M )0 = 0, D Tr M has no homogeneous elements in negative degrees. We
infer that (D Tr M )≥1 ' (DP1∗ )≥1 ' D Tr M (where for a module X ∈ gr Λ, (X)≥1
means X1 + X2 + · · · ). Set I = DP1∗ . Then D Tr M ' (I)≥1 = socn I for some
n.
Lemma 2.8. Let A be an indecomposable module in gr0 Λ and assume that A is
not an injective object in gr0 Λ.
(a) There exists an indecomposable nonprojective module M ∈ gr0 Λ such that
A ' h(D Tr M )0 i.
(b) If 0 → D Tr M → B → M → 0 is almost split in gr Λ, then there is an exact
sequence 0 → A → hB0 i → M → 0 which is almost split in gr0 Λ.
Proof. (a) Our assumption implies that A is not injective in gr Λ. Let E be an
injective envelope of A in gr Λ. We have the following exact sequence in gr Λ : 0 →
A0 → E0 → N → 0. We claim that N 6= 0. Otherwise we get an isomorphism
A ' hE0 i and this implies that there exists an indecomposable injective module E 0
such that hE00 i is contained in A. By applying 2.3 we obtain a contradiction to our
assumption on A, and therefore N 6= 0. Let F be an injective envelope of N in
gr Λ. The composition hE0 i → E0 → N → F can be extended to a map g : E → F .
We consider the induced map g 0 : (DE)∗ → (DF )∗ and its cokernel C ∈ gr0 Λ.
Since F is an injective envelope, g has no summand of the form 0 → F 0 for some
F 0 6= 0, so C has no nonzero projective summand. By dualizing we obtain an exact
g
sequence 0 → D Tr C q I → E −
→ F for some injective module I. By construction,
the image of the zero component of g is N = F0 , so we have an induced exact
sequence 0 → (D Tr C)0 q I0 → E0 → N → 0 at the degree zero level which means
that we have A0 = (D Tr C)0 q I0 . Thus A0 and (D Tr C)0 q I0 must generate
the same submodule of E, therefore A = hA0 i = h(D Tr C)0 i q hI0 i. Since A is
indecomposable, our assumption on A tells us that A = h(D Tr C)0 i. From 2.4
we infer that there is a unique indecomposable summand M of C which is not a
projective object in gr0 Λ, and we have A = h(D Tr M )0 i.
ON MODULES WITH LINEAR PRESENTATIONS
13
g
(b) Let 0 → D Tr M → B −
→ M → 0 be an almost split sequence in gr Λ. We
have the following commutative diagram.
0
/K
/ hB0 i
_
0
/ D Tr M
/B
g0
g
/M
/0
/M
/0
where g 0 = g|hB0 i . Then K is in gr0 Λ. This follows from general results on torsion
theories, as indicated earlier. For the convenience of the reader we explain why.
First we have that K0 6= 0 by 2.4. If K 6= hK0 i we have the commutative diagram
with exact rows:
/ hE0 i
/M
/0
/K
0
0
/ K/hK0 i
/W
0
0
h
/M
/0
where h : W → M is right almost split in gr0 Λ. Since K/hK0 i is generated in positive degree and M is not projective in gr0 Λ, we have a contradiction to Lemma 2.5.
Hence we have K = hK0 i.
It follows that K = D Tr M ∩ hB0 i = h(D Tr M )0 i, therefore K = A which
is indecomposable, and the map g 0 : hB0 i → M is in fact minimal right almost
split.
Note that we have proved for the torsion class T = gr0 Λ that when 0 → A →
B → C → 0 is almost split in gr Λ, with C in gr0 Λ, the sequence 0 → t(A) →
t(B) → C → 0 is almost split in gr0 Λ, so that we do not need to take summands.
As a consequence of Lemma 2.8 we have the following description of the projective objects in gr0 Λ:
Proposition 2.9. Let M be an indecomposable module in gr0 Λ. Then M is projective in gr0 Λ if and only if M is isomorphic to a module of the form Tr D(socn I)
for some indecomposable injective module I and some n > 0 or M is a projective
Λ-module.
Proof. Let M ∈ gr0 Λ be an indecomposable nonprojective Λ-module. Assume that
M is not a projective object in gr0 Λ. Then by 2.2, there exists an exact sequence
in gr0 Λ : 0 → A → B → M → 0 which is almost split in gr0 Λ. This means that A
is not an injective object in gr0 Λ and, by 2.8, A = h(D Tr M )0 i. We infer now from
2.3 that D Tr M cannot have the form socn I for some indecomposable injective
module I and some n > 0. The other direction is Lemma 2.7.
Remark. Let r 2 = 0 in Λ, and let X be the subset of gr Λ consisting of all the
simple Λ-modules and their shifts in gr Λ. Let FX ⊂ Ext1 (−, −) be the subfunctor
containing all exact sequences in gr Λ of the form 0 → A → B → C → 0 where, for
every S ∈ X we have an exact sequence 0 → Homgr Λ (C, S) → Homgr Λ (B, S) →
Homgr Λ (A, S) → 0. The exact sequences in the relative theory are those for which
A, B and C are all generated in one and the same degree, so we get a product
14
GREEN, MARTINEZ-VILLA, REITEN, SOLBERG, AND ZACHARIA
of the gri Λ, of which gr0 Λ is one component. It then follows from the results
of [ASo] that the simple and the injective modules are injective objects and that
the modules of the form Tr DS with S simple and the projective modules are the
projective objects, which fits with the result of 2.2, 2.3 and 2.9. When r 2 6= 0 the
relative injective modules and the relative projective modules are not related by
Tr D.
We turn now our attention to the category L(Λ).
Proposition 2.10. L(Λ) is contravariantly finite in gr0 Λ.
Proof. If M ∈ gr0 Λ we can construct a right L(Λ)-approximation of M in gr0 Λ
f
as follows: let P1 −
→ P0 → M → 0 be a minimal projective presentation of M in
gr Λ with P0 in gr0 Λ. Let Q1 be the largest summand of P1 which is generated in
degree one. We have the following commutative diagram.
Q1
_
P1
g
f
/ P0
/ P0
/M
αM
/M
/0
/0
where g = f |Q1 . We claim that the induced map αM : M → M is a right L(Λ)approximation of M : Suppose we have some X ∈ L(Λ) and a linear presentation
R1 → R0 → X → 0 and let h : X → M be in gr0 Λ. In the commutative diagram
R1
h1
P1
/ R0
h0
/ P0
/X
/0
h
/M
/0
it is easy to see that Im h1 ⊂ Q, so we have an induced map h0 : X → M such that
αM h0 = h.
One immediate consequence of Theorem 2.2 is the following:
Theorem 2.11. The category L(Λ) has almost split sequences.
Proof. Using the exact equivalence in 1.2 between L(Λ) and L(Λ/r 2 ) we see that an
f
g
exact sequence of graded modules and graded homomorphisms 0 → A −
→B−
→C →
f
g
0 is almost split in L(Λ) if and only if the induced sequence 0 → A −
→B−
→C →0
is almost split in L(Λ/r 2 ). But we have L(Λ/r 2 ) = gr0 (Λ/r 2 ) and we know from 2.2
that gr0 Λ has almost split sequences. This completes the proof.
We also have the following description of the projective and of the injective
objects in L(Λ).
Theorem 2.12. Let M be an indecomposable module in L(Λ).
(a) M is an injective object in L(Λ) if and only if M is simple or M/r 2 M is an
injective Λ/r2 -module.
(b) M is a projective object in L(Λ) if and only if M is a projective module or
M ' Tr DS for some simple Λ-module S.
ON MODULES WITH LINEAR PRESENTATIONS
15
Proof. (a) is immediate from 1.2, 1.3 and 2.3.
(b) Assume that M is nonprojective in gr Λ but M ' Tr DS for some simple
g
module S. Let 0 → S → E −
→ M → 0 be the almost split sequence ending at M
in gr Λ. Assume M is not a projective object in L(Λ). Then we have the following
commutative diagram with the top sequence nonsplit and exact in L(Λ):
0
/X
0
/S
/Y
/M
/0
/M
/ 0.
f
/E
g
If P1 → P0 → M → 0 is a linear presentation, then P1 /rP1 ' S, so S is generated
in degree one. Therefore Homgr Λ (X, S) = 0, thus f = 0, contradicting the fact
that g does not split. We conclude that M is a projective object in L(Λ).
For the other direction assume that M is a nonprojective module and that M
is projective in L(Λ). Then M/r 2 M is projective in gr0 (Λ/r2 ) = L(Λ/r2 ). From
2.5 it follows that there is an almost split sequence in gr(Λ/r 2 ) : 0 → S → E →
M/r2 M → 0 where E ∈ gr0 (Λ/r 2 ) and S is generated in degree one. It is also
clear that E has no simple summand and that we can write E = F/r 2 F for some
F ∈ L(Λ). We have the following exact commutative diagram:
0
/S
0
/S
α
/F
/M
/0
/E
/ M/r 2 M
/0
and we want to show that the top sequence is the almost split sequence of M in
gr Λ. The sequence clearly does not split since Homgr Λ (F, S) = 0 because F ∈ gr0 Λ
and S is generated in degree one. It is enough to show that α is left almost split in
gr Λ. Let f : S → X be a nonsplit monomorphism in gr Λ. Then f factors through
soc2 X, and, if soc2 X = X1 q · · · q Xn , then each induced map fi : S → Xi is not
an isomorphism. For each i, Xi ∈ gr0 (Λ/r2 ), and fi can be extended to E. Hence
composing with the map F → E, the map f : S ,→ soc2 X → X can be extended
to F , so α is left almost split. We conclude that M ' Tr DS.
3. Almost split sequences
In this section we continue our study of gr0 Λ and of L(Λ). In particular we
study the decomposability of the middle term of an almost split sequence 0 →
D Tr M → E → M → 0 for an indecomposable nonprojective module M in L(Λ)
or in gr0 Λ. We also show that if Λ is a selfinjective Koszul algebra, then for every
indecomposable nonprojective object M in KΛ, there exists an exact sequence
0 → A → B → M → 0 in K(Λ) which is almost split in K(Λ). But if M is an
indecomposable noninjective object in K(Λ), there need not exist in general an
exact sequence 0 → M → B → C → 0 in K(Λ) which is almost split in K(Λ).
Then in a sense, the category K(Λ) has right almost split sequences for graded
selfinjective Koszul algebras, but left almost split sequences do not exist in K(Λ)
for Loewy length greater than 3. Recall that Λ is assumed to be a finite dimensional
algebra over k.
We start with the following analysis of the indecomposable modules in L(Λ)
which are projective objects in gr0 Λ.
16
GREEN, MARTINEZ-VILLA, REITEN, SOLBERG, AND ZACHARIA
Theorem 3.1. Let M be an indecomposable nonprojective module such that M ∈
L(Λ). Then we have the following.
(i) M is a projective object in gr0 Λ if and only if D Tr M is simple.
(ii) Assume that M is not a projective object in gr0 Λ. Then there exists a
nonsplit exact sequence in gr0 Λ : 0 → soc2 D Tr M → B → M → 0 which is almost
split in the full subcategory C of gr Λ consisting of all graded modules X of the form
X = X≥0 (that is modules having no nonzero components in negative degrees).
Moreover, the induced sequence 0 → soc2 D Tr M → B/r2 M → M/r 2 M → 0 is
almost split over Λ/r 2 , in particular D TrΛ/r2 (M/r2 M ) ' soc2 D TrΛ M .
(iii) If M is a projective object in gr0 Λ, and if 0 → D Tr M → E → M → 0 is the
almost split sequence of M in gr Λ, then D Tr M is simple generated in degree one
and E ∈ gr0 Λ. In this case, the induced sequence 0 → D Tr M → E/r 2 E →
M/r2 M → 0 is almost split over Λ/r 2 , so in particular D TrΛ/r2 (M/r 2 M ) '
D Tr(M ).
Proof. (i) Let P1 → P0 → M → 0 be a minimal projective presentation of M
in gr Λ. Then P1 /rP1 ' soc D Tr M as graded modules, therefore, if D Tr M is
simple, it must be concentrated in degree one, i.e. D Tr M = (D Tr M )1 . Thus
(D Tr M )0 = 0 and it follows from 2.4 that M is a projective object in gr0 Λ. For
the other direction assume that D Tr M is not simple. Since M ∈ L(Λ) we have
D Tr M cogenerated in degree one, therefore (D Tr M )0 6= 0. Again, 2.4 implies
that M is not a projective object in gr0 Λ.
g
(ii) Let 0 → D Tr M → E −
→ M → 0 be almost split in gr Λ. Then the
g0
induced sequence 0 → (D Tr M )≥0 → E≥0 −→ M → 0 with g 0 = g|E≥0 is nonsplit in C, and, by the above discussion, we have soc2 D Tr M = (D Tr M )≥0 =
(D Tr M )0 + (D Tr M )1 , (and (D Tr M )0 6= 0). We show next that soc2 D Tr M is
indecomposable; this will imply that soc2 D Tr M ∈ gr0 Λ and, since gr0 Λ is closed
under extensions, we will also obtain that E≥0 = hE0 i ∈ gr0 Λ.
To prove that soc2 D Tr M is indecomposable, note that we have an exact sequence: P0∗ [1] → P1∗ [1] → Tr M [1] → 0 (where X[1] denotes the shift of X where
X[1]i = Xi+1 ). Therefore Tr M [1] ∈ L(Λop ) and by 1.3, Tr M [1]/ Tr M [1]r 2 is
indecomposable, hence Tr M/ Tr M · r 2 is indecomposable. But soc2 D Tr M '
D(Tr M/ Tr M · r 2 ) so it is also indecomposable as desired. Let B = hE0 i. We
g0
have a nonsplit exact sequence in gr0 Λ : 0 → soc2 D Tr M → B −→ M → 0 and
it is immediate that this sequence is almost split in C. By chopping off the degree
two parts and higher we have the following commutative diagram with exact rows,
where the bottom sequence is in gr0 (Λ/r2 ).
0
/ soc2 D Tr M
0
/ soc2 D Tr M
i
i
r2 B
_
∼
/ r2 M
_
/B
g0
/M
/0
/ M/r2 M
/0
/ B/r 2 B
g0
It is also immediate that the bottom sequence is nonsplit and, since M is in L(Λ),
both ends are indecomposable. Let X ∈ gr Λ be a module of Loewy length two
and let f : soc2 D Tr M → X be a nonsplit monomorphism. To show that i is
ON MODULES WITH LINEAR PRESENTATIONS
17
left almost split it suffices to assume that X ∈ C. Then f factors through i, so
there is a map h : B → X such that hi = f . Since X has Loewy length two,
h factors through B/r 2 B. This implies that i is left almost split in gr(Λ/r 2 ).
Therefore the bottom sequence is an almost split sequence in gr Λ/r 2 and we also
have D TrΛ/r 2 (M ) ' soc2 D TrΛ (M ).
(iii) By (i) D Tr M = S is simple. We have the following commutative diagram
where the bottom row is the almost split sequence of M in gr Λ and where P0 →
M → 0 is a projective cover of M in gr Λ with kernel L = ΩM .
/ P0
/M
/0
/L
0
0
/S
/E
g
/M
/0
The map L → S cannot be zero since g does not split. Since M ∈ L(Λ) it follows
that S is generated in degree one and that E ∈ gr0 Λ is a homomorphic image
of P0 . Again, as in part (ii), we chop off the degree two and higher parts to
obtain the following exact sequence in gr Λ/r 2 : 0 → S → E/r2 E → M/r 2 M →
0. To prove that this induced sequence is almost split in mod(Λ/r 2 ) and that
D TrΛ/r2 (M/r2 M ) ' D TrΛ M we proceed in the same fashion as in (ii).
Definition. Let Λ be a graded algebra. We say that a finitely presented module
M generated in (highest) degree b has a n-homogeneous presentation (n > 0), if
M has a minimal projective presentation P1 → P0 → M → 0 in gr Λ, with P0
generated in degree b and P1 generated in degree n + b.
Example. Examples of modules with n-homogeneous presentations abound; one
way to construct them is by looking at truncated path algebras of finite quivers
([BK]). Another easy example is the following. For q 6= 0 let
Λt,q = khx1 , x2 , . . . , xt i/hx2i , xi xj − qxj xi | i ≤ ji
be the quantized exterior algebra in t variables. Let M = Λ/r n Λ for some n < t.
Then M has a n-homogeneous presentation.
We have the following result on the decomposition of the middle term of the
almost split sequence of a module having a n-homogeneous presentation.
Proposition 3.2. Let M be an indecomposable nonprojective module, generated in
highest degree and having a n-homogeneous presentation. Let 0 → D Tr M → E →
M → 0 be its almost split sequence in gr Λ. Then either
(i) E is indecomposable or
(ii) E decomposes and there is an integer k with 0 ≤ k ≤ n such that r k M
has a simple summand and there exists an integer t with 0 ≤ t ≤ n such that
D Tr M/ soct D Tr M has a simple summand.
Proof. We may assume that M is generated in degree zero. Let P1 → P0 → M → 0
be a minimal projective presentation with P1 being generated in degree n. From
the exactness of P0∗ → P1∗ → Tr M → 0 it follows that soc D Tr M is all in degree
j
π
n, i.e. soc D Tr M = (D Tr M )n . Let 0 → D Tr M −
→E−
→ M → 0 be the almost
split sequence of M in gr Λ and assume
E decomposes as E = E1 q E2 with
that
E1 and E2 different from zero, so j = jj12 and π = (π1 , π2 ). It is well known that
the compositions h1 , h2 : D Tr M → M where hi = πi ji (i = 1, 2) are nonzero. Let
18
GREEN, MARTINEZ-VILLA, REITEN, SOLBERG, AND ZACHARIA
h denote h1 or h2 . If h(soc D Tr M ) 6= 0, then (M )n 6= 0 so r n M = M≥n , which is
generated in degree n, contains a semisimple submodule also generated in degree n.
This implies that r n M has a simple summand. Assume now that h(soc D Tr M ) =
h((D Tr M )n ) = 0. Since h 6= 0, there must be a k such that h((D Tr M )k ) 6= 0.
We choose 0 ≤ k ≤ n maximal with the property that h((D Tr M )≥k ) 6= 0 but
h((D Tr M )≥k+1 ) = 0. Again we see that h((D Tr M )≥k ) ⊂ r k M is semisimple so
rk M has a simple submodule in degree k and this submodule must split off. The
second statement follows using the same argument and the duality D: We first
note that (Tr M )[n] ∈ gr0 (Λop ) has a n-homogeneous presentation and its almost
split sequence is 0 → (DM )[n] → (DE)[n] → (Tr M )[n] → 0 with the middle term
decomposing. Then we use the duality and the first part.
It was proven in [BuRi] that if Λ is an arbitrary artin algebra and if for an
h
indecomposable module M the minimal projective presentation P1 −
→ P0 → M has
the property that P0 and P1 are both indecomposable and, if Im h 6⊆ r 2 P0 then
the almost split sequence 0 → D Tr M → E → M → 0 has indecomposable middle
term. Another way of constructing almost split sequences with indecomposable
middle term was studied in [K]. We have the following immediate consequence of
Proposition 3.2.
Corollary 3.3. Let M be an indecomposable graded module such that, up to a shift,
M is in L(Λ), and let 0 → D Tr M → E → M → 0 be its almost split sequence in
gr Λ. Then either
(i) E is indecomposable, or
(ii) E decomposes, M is simple or rM has a simple summand and D Tr M is
simple or D Tr M/ soc D Tr M has a simple summand.
Corollary 3.4. Let M be an indecomposable graded module such that up to shift
M is in L(Λ) and assume that the socle of M is in one and the same degree. Let
0 → D Tr M → E → M → 0 be the almost split sequence of M in gr Λ. If M has
Loewy length greater than two then E is indecomposable.
We now turn our attention to the study of almost split sequences for Koszul
modules over selfinjective algebras. We have the following application of 3.4 (see
also [Ri1]).
Proposition 3.5. Let Λ be an indecomposable finite dimensional graded selfinjective algebra with r 3 6= 0. Let M be an indecomposable nonprojective graded module
such that, up to shift, both M and ΩM have linear presentations, where ΩM denotes the first syzygy of M . Let 0 → D Tr M → E → M → 0 be the almost split
sequence of M in gr Λ. Then the middle term E is indecomposable. Moreover, if
N = ΩM , then the almost split sequence 0 → D Tr N → F → N → 0 in gr Λ has
indecomposable middle term. Furthermore, N is not a projective object in gr0 Λ.
Proof. We observe first that all the indecomposable projective modules have the
same Loewy length by [M 1], and by assumption we have that N = ΩM has Loewy
length n − 1 where n = LL(Λ). Therefore, since r 3 6= 0, we have LL(N ) > 2.
The module N is a submodule of a projective injective module generated in highest
degree, so soc N is all in one and the same degree. The module N = ΩM is in
L(Λ), so we may apply 3.4 to obtain the indecomposability of F . Next, we apply
Ω−1 to the exact sequence 0 → D Tr N → F → N → 0. We obtain that the
almost split sequence of M has the form: 0 → Ω−1 D Tr N → Ω−1 F q P → M → 0
ON MODULES WITH LINEAR PRESENTATIONS
19
where P is zero or projective. If P 6= 0, then M ' P/ soc P , but such a module
cannot have a linear presentation unless r 2 = 0. Therefore P = 0 and the almost
split sequence of M is 0 → Ω−1 D Tr N → Ω−1 F → M → 0 and E ' Ω−1 F is
indecomposable. It remains to show that no shift in gr Λ of N is a projective object
in gr0 Λ. In view of 3.1 it is enough to prove that D Tr N is not simple. This is clear
since D Tr N ' νΩ2 N where ν is the Nakayama functor, has same Loewy length
as Ω2 N , hence LL(D Tr N ) = n − 1 and n > 3.
We turn our attention to the question whether almost split sequences exist in
the category K(Λ). We have the following result:
Theorem 3.6. Let Λ be an indecomposable selfinjective Koszul algebra, and let
M ∈ K(Λ) be an indecomposable nonprojective module. There exists an almost
split sequence in K(Λ) : 0 → soc2 D Tr M → B → M → 0.
Proof. If r 2 = 0, the claim follows since K(Λ) = L(Λ) = gr0 Λ in this case. If r 2 6= 0,
D Tr M cannot be simple, therefore M is not a projective object in gr0 Λ and by 3.1
we also have the following nonsplit exact sequence in gr0 Λ which is almost split in C:
0 → soc2 D Tr M → B → M → 0. We also have soc2 D Tr M = rn−2 D Tr M . We
know that after shifting D Tr M is Koszul; then we get by [GM1] that soc2 D Tr M
is Koszul. Since K(Λ) is closed under extensions, it follows that B ∈ K(Λ). The
sequence being almost split in gr0 Λ implies that it is also almost split in K(Λ).
We show now that left almost split sequences need not always exist in the category of Koszul modules, even for selfinjective algebras.
Theorem 3.7. Let Λ be an indecomposable selfinjective Koszul algebra such that
r3 6= 0. There exists an indecomposable Koszul module N which is not an injective
object in K(Λ), for which there is no almost split sequence of the form 0 → N →
B → M → 0 in K(Λ).
Proof. Let n > 3 be the Loewy length of Λ. Let S be a simple Λ-module and let
N = Ω2 S be its second syzygy. Clearly up to shift N ∈ K(Λ) and, if 0 → N → P1 →
P0 → S → 0 is exact with P1 → P0 → S being a (minimal) linear presentation of
S, then we have an induced nonsplit exact sequence 0 → N → rP1 → r 2 P0 → 0
(the map N → rP1 is not an isomorphism since r 2 P0 /r3 P0 6= 0). Therefore, since
rP1 and r2 P0 are again Koszul modules, up to shift, (by [GM1]) generated in the
same degree as N , we conclude that N is not an injective object in K(Λ). Suppose
that we have an almost split sequence in K(Λ) of the form 0 → N → B → M → 0.
Then, by Theorem 3.6 we must have N ' soc2 D Tr M . But N = Ω2 S has Loewy
length n − 1 > 2, which yields a contradiction.
We conclude this section by observing that there is no shortage of selfinjective
Koszul algebras, for instance the algebras Λt,q in the example after Theorem 3.1.
(See [M2] for a more detailed treatment.)
4. Appendix. The finitistic projective dimension for Koszul modules
We show in this section that if Λ = Λ0 +Λ1 +· · ·+Λr is a finite dimensional graded
algebra, then the finitistic projective dimension of the Koszul modules is finite, and
if Λ is a Koszul algebra then either gl. dim. Λ = ∞ or gl. dim. Λ ≤ n(r+1) where n is
the number of nonisomorphic simple Λ-modules. Recall that the finitistic projective
dimension of a category A ⊆ gr Λ is sup{pd M | M in A such that pd M < ∞}.
20
GREEN, MARTINEZ-VILLA, REITEN, SOLBERG, AND ZACHARIA
We use the results of [Z] in this section and we will include some of them for the
readers convenience.
Definition. Let C be the full subcategory of gr Λ consisting of those graded modules without component in negative degrees. Let M ∈ C and let P ∗ be a minimal
projective resolution of M in gr Λ (thus also in C).
(P ∗ ) : · · · → Pn → Pn−1 → · · · → P1 → P0 → M → 0
P
For each i ≥ 0 define ai = j≥0 (−1)j dimk Extjgr Λ (M, Λ0 [i]), where Λ0 [i] denotes
P
the i-th shift of Λ0 in gr Λ. Next we define the series of M , s(M ) = i≥0 ai xi ∈
Z[[x]]. It is easy to see that s(M ) does not depend on the finitely generated resolution of M .
We recall the following results from [Z].
Lemma 4.1. (i) Let 0 → A → B → C → 0 be a short exact sequence in C. Then
s(A) + s(C) = s(B).
(ii) For each M ∈ C, s(M ) is a rational function [W ].
Definition. We define the measure µ(M ) of a module M ∈ C to be µ(M ) =
deg s(M ), the degree of the numerator minus the degree of the denominator of the
rational function s(M ).
The proof of the following lemma is immediate.
Lemma 4.2. (i) Let M ∈ C. For each integer i such that M [i] ∈ C we have
s(M [i]) = xi s(M ). Therefore µ(M [i]) = µ(M ) + i.
(ii) Let s1 and s2 be two rational functions.
Then deg(s1 + s2 ) ≤
max{deg s1 , deg s2 }.
Lemma 4.3. Let Λ = Λ0 + · · · + Λr and let M = M0 + M1 + · · · + Ms ∈ C be
indecomposable. Then we have the inequality µ(M ) ≤ s + 1 + µ(Λ 0 ). In particular,
if M ∈ gr0 Λ then µ(M ) ≤ r + 1 + µ(Λ0 ) and thus {µ(M ) | M ∈ gr0 Λ} is bounded.
Proof. We proceed by induction on the graded length s of M . We have a short
exact sequence in C : 0 → M 0 → M → M0 → 0 where M 0 = M1 + · · · + Ms .
Therefore, since s(M ) = s(M 0 ) + s(M0 ), we get from Lemma 4.2 that µ(M ) ≤
max{µ(M0 ), µ(M 0 )} = max{µ(M0 ), µ(M 0 [−1]) + 1}. Then we apply the induction
hypothesis.
Definition. We say that a finitely generated module generated in (highest) degree
b has a homogeneous projective resolution if there exists a sequence of integers
0 = n0 < n1 < n2 < · · · < nk < · · · and a projective resolution of M in gr Λ : · · · →
Pk → Pk−1 → · · · → P1 → P0 → M → 0 such that for each i, Pi is generated in
degree b + ni . We denote by H(Λ) ⊂ gr Λ the full subcategory of gr Λ consisting of
those modules having homogeneous linear resolutions. It is clear that K(Λ) ⊆ H(Λ).
Lemma 4.4. Let M ∈ gr0 Λ and assume also that M ∈ H(Λ). If M has finite
projective dimension we have pd M ≤ µ(M ), with equality occurring if and only if
M is a Koszul module.
Proof. If pd M < ∞ then s(M ) is a polynomial and, if 0 → Pk → · · · → P0 →
M → 0 a projective resolution of M in gr Λ, we have µ(M ) = nk ≥ k. The rest
follows trivially.
ON MODULES WITH LINEAR PRESENTATIONS
21
Putting 4.3 and 4.4 together we get the following.
Theorem 4.5. Let Λ be a finite dimensional graded k-algebra. Then the finitistic
projective dimension of H(Λ) (and therefore of K(Λ)) is finite.
The following result has the same proof as Theorem 2.13 in [Z].
Theorem 4.6. Let Λ = Λ0 + Λ1 + · · · + Λr be a graded finite dimensional algebra
such that Λ0 has a linear resolution (for instance Λ could be a Koszul algebra).
Then, either gl. dim. Λ = ∞ or gl. dim. Λ ≤ n(r + 1) = n · LL(Λ) where n is the
number of nonisomorphic simple modules.
References
[ARS] Auslander, M., Reiten, I. and Smalø, S: Representation Theory of Artin Algebras, Cambridge studies in advanced mathematics, vol 36, 1995.
[ASm] Auslander, M. and Smalø, S.: Almost split sequences in subcategories, Journal of Algebra 69 (2), 1981, 426–454.
[ASo] Auslander, M. and Solberg, O.: Relative homology and representation theory I, Communications in Algebra, 21 (9), 1993, 2995–3031.
[BGS] Beilinson, A., Ginsberg, V. and Soergel, W.: Koszul duality patterns in representation
theory, Journal of the American Mathematical Society, Vol. 9, No. 2 (1996) 473–528.
[BK]
Butler, M. C. R. and King, A.: Minimal resolutions of algebras, preprint, 1996.
[BuRi] Butler, M. C. R. and Ringel, C. M.: Auslander-Reiten sequences with few middle terms
and applications to string algebras, Communications in Algebra, 15 (1987), 147–179.
[GoGr] Gordon, R. and Green, E. L.: Representation theory of graded artin algebras, Journal of
Algebra, Vol. 76, No. 1 (1982).
[GM1] Green, E. L. and Martinez-Villa, Roberto: Koszul and Yoneda algebras, Canadian Mathematical Society Conference Proceedings, vol. 18 (1996), 247–297.
[GM2] Green, E. L. and Martinez-Villa, Roberto: Koszul and Yoneda algebras II, preprint, 1996.
[GZ]
Green, E. L. and Zacharia, D.: The cohomology ring of a monomial algebra, Manuscripta
Mathematica, 85, 1994, 11–23.
[K]
Krause, H.: The kernel of an irreducible map, Proceedings of the American Mathematical
Society, vol 121, Number 1, 1994, 57–66.
[M1]
Martinez-Villa, Roberto: Graded, selfinjective and Koszul algebras, Preprint, 1996.
[M2]
Martinez-Villa, Roberto: Contravariantly finite categories and torsion theories, Preprint,
1995.
[Ri1]
Ringel, C. M.: Cones, Canadian Mathematical Society Conference Proceedings, vol 18,
1996, 583–586.
[Ri2]
Ringel, C. M.: The Liu-Schulz example, Canadian Mathematical Society Conference Proceedings, vol. 18, 1996, 587–600.
[Sm]
Smith, S. P.: Some finite dimensional algebras related to elliptic curves, Representation
theory of algebras and related topics, Proceedings of the Workshop, Canadian Math. Soc.
Conf. Proc., Vol. 19, 1996, 315–348.
[St]
Stenstrom, B.: Rings of Quotients, Springer, Berlin, 1975.
[W]
Wilson, G.: The Cartan map on categories of graded modules, Journal of Algebra 85
(1983), 390–398.
[Z]
Zacharia, D.: Graded artin algebras, rational series, and bounds for homological dimensions, Journal of Algebra, vol. 106, 1987, 476–483.
22
GREEN, MARTINEZ-VILLA, REITEN, SOLBERG, AND ZACHARIA
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA
E-mail address: [email protected]
Instituto de Matematicas, UNAM, Ciudad Universitaria, 04510 Mexico, D.F., Mexico
E-mail address: [email protected]
Institutt for matematiske fag, NTNU, Lade, N–7034 Trondheim, Norway
E-mail address: [email protected]
Institutt for matematiske fag, NTNU, Lade, N–7034 Trondheim, Norway
E-mail address: [email protected]
Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA
E-mail address: [email protected]